## 231 Compressor and turbine efficiency

The work output from (or input to) a turbomachine can be found from the first law of thermodynamics. From this law the steady

flow energy equation may be derived. A turbomachine has one inlet and one outlet port. The steady flow energy equation becomes:

where Q = heat transfer rate (+ ve to the system);

W = work transfer rate (+ ve by the system); m = mass flow rate; h = specific enthalpy; KE = specific kinetic energy; PE - specific potential energy; suffixes 1,2 = inlet and outlet ports respectively.

Denoting the stagnation enthalpy (h0) as h0 = h + KE (2.2)

and neglecting changes in potential energy and heat transfer, since these terms are small, this becomes ds > dQ/T

The second law can also be used to show that ideal adiabatic compression or expansion takes place at constant entropy.

One definition of the efficiency of a compressor is the power required for ideal, adiabatic compression divided by the actual power required in a non-ideal, non-adiabatic compressor, working with the same inlet pressure and temperature and outlet pressure.

isentropic power actual power

Hence i)ts is termed the isentropic efficiency of the compressor. From eqns (2.3), (2.6) and (2.8),

Both air and exhaust gas are considered as perfect gases. Hence they obey the equation of state.

where P,v,R and 7" denote pressure (absolute), specific volume, gas constant and temperature respectively. The specific heat capacity at constant pressure (Cp) for a perfect gas is given by:

where suffices 'is' and 'TT' denote 'isentropic' and 'total to total', meaning an efficiency based on total temperature values. Note that the work required by a non-ideal compressor exceeds that of an isentropic compressor, hence the exit air temperature To2 is higher than To2js.

For isentropic compression, pressure and temperature are related by the expression

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